Universal.Kriging {rkriging}

Universal Kriging

Description

This functions fits the universal kriging model to the data.

Usage

Universal.Kriging(
  X,
  y,
  basis.function,
  interpolation = TRUE,
  fit = TRUE,
  kernel = NULL,
  kernel.parameters = list(),
  nlopt.parameters = list()
)

Arguments

X

a matrix for input (feature)

y

a vector for output (target), only one-dimensional output is supported

basis.function

the basis functions for specifying the prior mean

interpolation

interpolation whether to interpolate, for noisy data please set interpolate=FALSE

fit

whether to fit the length scale parameters from data

kernel

a kernel class object

kernel.parameters

a list of parameters required for the kernel, if no kernel class object is provided

nlopt.parameters

a list of parameters required for NLopt, including choice of optimization algorithm and maximum number of evaluation

Details

Universal kriging permits a more general function of mean, which can be specified using basis.function. Please see Santner et al. (2003) for details.

For data from deterministic computer experiments, use interpolation=TRUE and will give an interpolator. For noisy data, use interpolation=FALSE, which will give an approximator of the underlying function.

The kernel choices are required and can be specified by (i) providing the kernel class object to kernel or (ii) specifying the kernel type and other parameters in kernel.parameters. Please see examples section of Fit.Kriging for detail usages.

When the lengthscale / correlation parameters are unknown, all parameters including the constant mean can be estimated via Maximum Likelihood method by setting fit=TRUE. The initial / lower bound / upper bound of the lengthscale parameters can be provided in kernel.parameters, otherwise a good initial and range would be estimated from the data. The optimization is performed via NLopt, a open-source library for nonlinear optimization. All gradient-free optimization methods in NLopt are supported and can be specified in nlopt.parameters. See nloptr::nloptr.print.options() for the list of available derivative-free algorithms (prefix with NLOPT_GN or NLOPT_LN). The maximum number of optimization steps can also be defined in nlopt.parameters. Please see examples section of Fit.Kriging for detail usages.

Value

A Universal Kriging Class Object.

Author(s)

Chaofan Huang and V. Roshan Joseph

References

Santner, T. J., Williams, B. J., Notz, W. I., & Williams, B. J. (2003). The design and analysis of computer experiments (Vol. 1). New York: Springer.

See Also

Fit.Kriging, Predict.Kriging, Get.Kriging.Parameters.

Examples

# one dimensional example 
f <- function(x) {
  x <- 0.5 + 2*x
  y <- sin(10*pi*x)/(2*x) + (x-1)^4
  return (y)
}

set.seed(1234)
# train set
n <- 30
p <- 1
X <- matrix(runif(n),ncol=p)
y <- apply(X, 1, f)
newX <- matrix(seq(0,1,length=1001), ncol=p)

basis.function <- function(x) {c(1,x[1],x[1]^2)}

# approach 1
kriging <- Universal.Kriging(X, y, basis.function=basis.function,
                             interpolation=TRUE, fit=TRUE, 
                             kernel.parameters=list(type="Gaussian"))
pred <- Predict.Kriging(kriging, newX)
plot(newX, f(newX), "l")
points(X, y, pch=16, col="blue")
lines(newX, pred$mean, col="red", lty=2)
lines(newX, pred$mean-2*pred$sd, col="red", lty=3)
lines(newX, pred$mean+2*pred$sd, col="red", lty=3)
Get.Kriging.Parameters(kriging)

# approach 2
kriging <- Fit.Kriging(X, y, interpolation=TRUE, fit=TRUE, model="UK",
                       model.parameters=list(basis.function=basis.function),
                       kernel.parameters=list(type="Gaussian"))
pred <- Predict.Kriging(kriging, newX)
plot(newX, f(newX), "l")
points(X, y, pch=16, col="blue")
lines(newX, pred$mean, col="red", lty=2)
lines(newX, pred$mean-2*pred$sd, col="red", lty=3)
lines(newX, pred$mean+2*pred$sd, col="red", lty=3)
Get.Kriging.Parameters(kriging)